[seqfan] Re: how to simplify sin(3*Pi/10) etc in Maple

[Brendan McKay]

simplify(convert(sin(3*Pi/10),exp));  works for this example.

In general it could be a zero of an irreducible polynomial of high degree.

Consider a more difficult case:  y = sin(x) where x = 3*Pi/11.
We know that sin(11*x)=0, so expand sin(11*x) in terms of y and factor:

expand(sin(11*x)):
simplify(%,{cos(x)^2=1-sin(x)^2}):
P := factor(subs(sin(x)=y,%));
      -y*(1024*y^10 - 2816*y^8 + 2816*y^6 - 1232*y^4 + 220*y^2 - 11)
Now note that sin(3*Pi/11) is a root of the larger factor.

Sometimes, annoyingly, the second line gives a polynomial in cos(x)
instead of sin(x).  I don't know a simple way to correct that.

Alternatively note that sin and cos of a rational multiple of Pi is a
root of some Chebyshev polynomial, which Maple knows about.

Brendan.

On 25/5/2023 10:33 am, Neil Sloane wrote:
> How do I get Maple to evaluate expressions like sin(3*Pi/10) ?
>
> That one is easy, it is (1+sqrt(5))/4, just look up the decimal expansion
> in the OEIS and you get the answer, A019863.
>
> But in general? I can't find an answer in any of the help pages.
>
> If you know the answer please post it here, to avoid duplication of effort.
>
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